I need to proof that the Pareto random variable is a mixture of the Gamma and Exponential distribution but need help with one part of the proof.
Consider $X$ being Exponential with parameter $\lambda$ and $\Lambda$ being Gamma with parameters $\alpha$ and $\beta$. So we can say the mixture distribution of $X$ is
$$
\begin{align}
f_{X|\alpha, \beta} &= \int_0^\infty \lambda e^{-\lambda x} \cdot \frac{1}{\Gamma (\alpha) \beta ^ \alpha} \lambda^{\alpha – 1} e^{-\frac \lambda \beta} d\lambda\\
&=\frac {1} {\Gamma (\alpha) \beta ^ \alpha} \int_0^\infty \lambda ^\alpha e^{-\lambda x – \lambda \frac 1 \beta} d\lambda\\
&=\frac{\Gamma (\alpha +1)}{\Gamma (\alpha) \beta^\alpha} \int_0^\infty \frac{\lambda ^\alpha e^{-\lambda x – \lambda \frac 1 \beta}}{e^{-\lambda}\lambda^\alpha} d\lambda\\&=
\frac{\alpha}{\beta^\alpha}\int_0^\infty e^{\lambda \frac{\beta – x -1}{\beta}} d\lambda\\
&=\frac{\alpha}{\beta^\alpha}\begin{bmatrix} \frac{\beta}{\beta – x -1}e^{\lambda\frac{\beta – x – 1}{\beta}} d\lambda\end{bmatrix}^\infty _0\\&=?\\
&=\frac{\alpha}{\beta^\alpha}\begin{bmatrix}\frac{\beta}{\beta x+1}\end{bmatrix}^{\alpha +1}
\end{align}
$$
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compound of gamma and exponential distribution